math.DSMar 17, 2026

Data-driven forced response analysis with min-max representations of nonlinear restoring forces

Authors: Akira Saito, Hiromu Fujita

Organizations: Department of Mechanical Engineering, Meiji University, Kawasaki, Kanagawa 214-8571, Japan.

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Abstract

This paper discusses a novel data-driven nonlinearity identification method for mechanical systems with nonlinear restoring forces such as polynomial, piecewise-linear, and general displacement-dependent nonlinearities. The proposed method is built upon the universal approximation theorem that states that a nonlinear function can be approximated by a linear combination of activation functions in artificial neural network framework. The proposed approach utilizes piecewise linear springs with initial gaps to act as the activation functions of the neurons of artificial neural networks. A library of piecewise linear springs with initial gaps are constructed, and the contributions of the springs on the nonlinear restoring force are determined by solving the linear regression problems. The piecewise linear springs are realized by combinations of min and max functions with biases. The proposed method is applied to a Duffing oscillator with cubic stiffness, and a piecewise linear oscillator with a gap and their nonlinearities are successfully determined from their free responses. The obtained models are then used for conducting forced response analysis and the results match well with those of the original system. The method is then applied to experimentally-obtained free response data of a cantilevered plate that is subjected to magnetic restoring force, and successfully finds the piecewise linear representation of the magnetic force. It is also shown that the obtained model is capable of accurately capturing the steady-state response of the system subject to harmonic base excitation.

Figures & tables

Figure 1: Schematic diagram of a single layer perceptron
Figure 2: Schematic diagram of multiple piecewise springs acting on the mass
Figure 3: Convergence of the approximated function with respect to the number of functions. p2=10p_{2}=10 . The regression was conducted for data in the filled region ( −5⩽x⩽5-5\leqslant x\leqslant 5 ).
Figure 4: Spring constants versus the initial gap values
Figure 5: Time history of the oscillator with the original cubic nonlinearity and with the min-max representations
Figure 6: Comparison of the time-frequency relationships using scalogram of the response
(M,N)(M,N)(8,8)(8,8)(32,32)(32,32)(128,128)(128,128)
RMSE (m)2.382.380.4970.4970.2150.215
Table 1: RMSE values for the approximations
Figure 7: Comparison of the forced response results with single cubic stiffness and multiple PWL springs, along with classical approximate solutions and backbone curve
Figure 8: Schematic diagram of the piecewise linear system
Figure 9: Convergence of the approximated function with respect to the number of functions. p2=10p_{2}=10 . The regression was conducted for data in the filled region ( −5⩽x⩽5-5\leqslant x\leqslant 5 ).
Figure 10: Spring constants versus the initial gap values for the piecewise linear stiffness ( L=0.5L=0.5 )
Figure 11: Comparison of the free response results with a single piecewise linear stiffness and multiple PWL stiffnesses.
(M,N)(M,N)(8,8)(8,8)(32,32)(32,32)(256,256)(256,256)
RMSE (m)0.5540.5540.2290.2290.0280.028
Table 2: RMSE values for the approximations
Figure 12: Comparison of the forced response results with a single piecewise linear stiffness and multiple PWL springs
Figure 13: Phase portraits and Fourier spectra at primary and secondary resonances
Figure 14: Test equipment for free vibration test. The plate is in equilibrium state.
Figure 15: Time response of the displacement of the plate tip and backbone curve obtained from the free response
Figure 16: Identified potential and its associated magnetic force
Figure 17: Spring constants versus the initial gap values for the min-max functions
Figure 18: Results of free response ( M=32M=32 ). The regression was conducted for data in the filled region ( 0⩽t⩽30\leqslant t\leqslant 3 ).
Figure 19: Convergence study with respect to the number of ϕ(x,gi)\phi(x,g_{i}) (Fit duration=50%)
Figure 20: Convergence study with respect to the fit duration to total duration ( M=32M=32 )
Figure 21: Test equipment for forced vibration test
Figure 22: Comparison of the measured forced response time histories and those computed with the proposed approach at 15Hz.
Figure 23: Comparison of the measured forced response results and those computed with the proposed approach
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